\section{Background}
\label{sec:background}
%% This section provides the basic background for computing the minimum
%% transmitting power for on-chip wireless communication in order to
%% guarantee a given bit error rate (BER). Then, Friis transmission
%% equation is introduced in order to estimate the fraction of the
%% overall transmitting power that reaches the receiving antenna.
Given a transmitting and a receiving antenna, this section provides 
the background on how computing the minimum transmitting power which
guarantees a certain data rate and a maximum bit error rate (BER).

%------------------------------------------------------------------------------

\subsection{Signal Strength Requirements}
\label{sec:ssreq}
For adapting the baseband signal to the wireless medium, the most used
modulation scheme in the WiNoC context is Amplitude Shift Keying or On
Off Keying (ASK-OOK)~\cite{ditommaso_hoti11, deb_tc13, dt_14}.  The
reliability of the ASK-OOK modulation (in terms of BER) is related to
the energy spent per bit, $E_{bit}$, as follows:
\begin{equation}
  BER=Q\bigg( \sqrt{\frac{E_{bit}}{N_0}}\bigg),
  \label{eq:ber}
\end{equation}
where $N_0$ is the transceiver noise spectral density (noise
introduced by the transceiver) and the $Q$ function is the tail
probability of the standard normal distribution which is defined in
Eqn.~(\ref{eq:q_func}).
\begin{equation} 
  \label{eq:q_func}
  Q(x)=\frac{1}{\sqrt{2\pi}}\int_{x}^{\infty} e^{-\frac{y^2}{2}}dy   
\end{equation}
Since $E_{bit}=P_r/R_b$, where $P_r$ is the power received at the terminal
of the receiver antenna while $R_b$ is the data rate, the required
transmitting power for a given data rate and BER requirement and for a
given transceiver's thermal noise can be computed as:
\begin{equation}
  P_r = E_{bit} \cdot R_b = \left[Q^{-1}(BER)\right]^2 N_0 R_b,
  \label{eq:pr}
\end{equation}
where $Q^{-1}$ is the inverse of the $Q$ function. Thus, the minimum
transmitting power needed for reaching the receiving antenna can be
computed as:
\begin{equation}
  P_t = P_r / G_a,
  \label{eq:pt}
\end{equation}
where, $P_r$ is given by Eqn.~(\ref{eq:pr}) and $G_a$ is the
attenuation introduced by the wireless medium ($G_a<1$). The next
subsection describes how the attenuation $G_a$ can be computed.

%------------------------------------------------------------------------------

\subsection{Wireless Medium Attenuation}
\label{ssec:friis}
\begin{figure}
  \centering
  \includegraphics[width=0.4\textwidth]{pictures/friis.eps}
  \caption{Friis transmission equation: geometrical orientation of
    transmitting and receiving antennas. As indicated, considering a
    spherical coordinate system, $\phi$ is the azimuthal angle in the
    XY plane, where the X-axis is $0^\circ$ and Y-axis is
    $90^\circ$. $\theta$ is the elevation angle where the Z-axis is
    $0^\circ$, and the XY plane is $90^\circ$.}
  \label{fig:friis}
\end{figure}
As discussed in the previous subsection, the required transmitting
power depends on several factors, including, the type of modulation,
the transceiver noise figure, and the attenuation introduced by the
wireless medium. Let us consider Fig.~\ref{fig:friis} which shows a
transmitting antenna with an output power $P_t$ and a relative angle
respect the receiving antenna of $(\theta_t,\phi_t)$, and a receiving
antenna, located at distance $R$, with a relative angle respect the
transmitting antenna of $(\theta_r,\phi_r)$. The fraction of the
transmitting power that reaches the terminal of the receiving antenna,
$P_r$, can be computed by means of the Friis transmission
equation~\cite{balanis2008modern} valid when $R>2D^2/\lambda$, where
$D$ is the the maximum dimension of antenna (axial length in the
considered case) and $\lambda$ is the wavelength. The Friis
transmission equation is:
\begin{equation}
  G_a = \frac{P_r}{P_t} = e_t e_r \frac{\lambda^2 D_t(\theta_t,\phi_t)
  D_r(\theta_r,\phi_r)}{(4\pi R)^2},
  \label{eq:friis_complex}
\end{equation} 
where:
\begin{itemize}
  \item $e_t$ and $e_r$ are the efficiencies of the transmitting and
    receiving antenna, respectively. These parameters mainly represent
    the signal losses in the silicon substrate. For reducing such
    contribution, high resistivity Silicon on Insulator (SoI)
    substrates ($>1~\mathrm{K \Omega cm}$) can be
    used~\cite{montusclat_ecwt05} or a polyamide stratus (few micron
    thick) can be inserted under the antenna~\cite{lee_mobicom_09}.
  
  \item $D_t$ and $D_r$ are the directivities of the transmitting and
    receiving antenna, respectively. They quantify how much better the
    antenna can transmit or receive in a specific direction.
  
  \item $\lambda$ is the effective wavelength. For an IC substrate, it
    is estimated by using the material properties of the top IC layers
    (silicon dioxide $\epsilon_r=3.9$)~\cite{gutierez_jsac09}.
\end{itemize}
Eqn.~(\ref{eq:friis_complex}) highlights the parameters which
determine the gain $G_a$ and represents a first order model of the
wireless channel. In fact, second order effects such as, polarization
matching, wave reflections, and multi-path effects are not modelled by
Eqn.~(\ref{eq:friis_complex}).  However, in practical cases, $G_a$
computation is estimated by means of Eqn.~(\ref{eq:friis_measured})
\begin{equation}
  G_a=\frac{P_r}{P_t}=\frac{|S_{12}|}{(1-|S_{11}|)(1-|S_{22}|)},
  \label{eq:friis_measured}
\end{equation}
where, $S_{11}$, $S_{12}$, and $S_{22}$ are the scattering
parameters. Such parameters are gathered by using field solver
simulation tools~\cite{floyd_jssc02} or by direct measures from
realized prototypes.

%------------------------------------------------------------------------------

%% \subsection{WiNoC Transceiver Energy Model}
%% Once that attenuation introduced by the wireless medium for a certain
%% communication have been computed as described above, Eqn.~\ref{eq:pt}
%% can be used to compute the required transmitting power in order to
%% guarantee a certain level reliability constrain. Thus, the main scope
%% of this section is now the introduction of a model useful for computing
%% the energy consumption for a generic WiNoC. While the computation of 
%% energy contribution due to electrical links and switches is well known,
%% the contributes due to radio front-end have been only recently
%% investigated. The energy consumed by the wireless front-end depends by several 
%% factors such as transceiver architecture, wireless medium attenuation,
%% kind of modulation and datarate. Unlike digital circuitry, analog 
%% devices such as oscillators amplifiers consumes a huge amount of 
%% energy also when the transceiver is not involved in a data transfer.
%% This contribute is mainly due to the bias current flowing in any 
%% signal amplifier and oscillator. For these reasons, power 
%% contribution can be divided in two main terms namely, dynamic e static 
%% power consumption. 
%% In particular, the average transmitter or receiver power consumption 
%% can be expressed as:
%% \begin{equation}
%%   P^{tx/rx}_{tot}=P^{tx/rx}_{dyn} + P^{tx/rx}_{static},
%%   \label{eq:energy_winoc}
%% \end{equation}
%% where $P^{tx}_{dyn}(i,j)$ and $P^{tx}_{static}$ are the 
%% dynamic and static power consumption respectively. 
%% Introducing the transmitter efficiency $\eta=P_t/P^{tx}_{dyn}$, 
%% the energy spend per bit, independently by static contributes
%% (which consumes power also when single bit does not transfered), can be 
%% expressed as proposed in~\cite{Abadal_tnet14}:
%% \begin{equation}
%%   E(i,j)=\frac{P_t(i,j)/ \eta + N \cdot P^{rx}_{dyn}}{R_b},
%%   \label{eq:energy_winoc}
%% \end{equation}
%% where $P_t(i,j)$ is the actual transmitted power as expressed in Eqn.~\ref{eq:pt}
%% which can depends by a specific $<source,destination>$ if the tunable transmitting power
%% mechanism~\cite{mineo_date14, mineo_date15} is applied. 
%% The scope of the proposed technique is to reduce only the dynamic energy consumption  as expressed in the Eqn.~\ref{eq:energy_winoc}. The static energy contribute should be added to the leakage energy due to electrical switches and links for obtaining the overall energy consumption.
%% Unless otherwise specified in the rest of the paper we refer to $E$ for indicating the dynamic
%% transceiver energy contribution. 
